# Properties

 Label 504.2.bl Level 504 Weight 2 Character orbit bl Rep. character $$\chi_{504}(17,\cdot)$$ Character field $$\Q(\zeta_{6})$$ Dimension 16 Newform subspaces 1 Sturm bound 192 Trace bound 0

# Related objects

## Defining parameters

 Level: $$N$$ = $$504 = 2^{3} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 504.bl (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ = $$21$$ Character field: $$\Q(\zeta_{6})$$ Newform subspaces: $$1$$ Sturm bound: $$192$$ Trace bound: $$0$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(504, [\chi])$$.

Total New Old
Modular forms 224 16 208
Cusp forms 160 16 144
Eisenstein series 64 0 64

## Trace form

 $$16q - 8q^{7} + O(q^{10})$$ $$16q - 8q^{7} + 12q^{19} + 12q^{25} + 24q^{31} + 4q^{37} + 8q^{43} + 32q^{49} - 28q^{67} - 60q^{73} - 32q^{79} - 32q^{85} - 84q^{91} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(504, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
504.2.bl.a $$16$$ $$4.024$$ $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ None $$0$$ $$0$$ $$0$$ $$-8$$ $$q+(\beta _{4}-\beta _{10})q^{5}+(-\beta _{3}-\beta _{9})q^{7}+(-\beta _{2}+\cdots)q^{11}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(504, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(504, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(21, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(42, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(63, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(84, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(126, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(168, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(252, [\chi])$$$$^{\oplus 2}$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 
$3$ 
$5$ $$1 - 26 T^{2} + 343 T^{4} - 3142 T^{6} + 22997 T^{8} - 145796 T^{10} + 840842 T^{12} - 4532272 T^{14} + 23186506 T^{16} - 113306800 T^{18} + 525526250 T^{20} - 2278062500 T^{22} + 8983203125 T^{24} - 30683593750 T^{26} + 83740234375 T^{28} - 158691406250 T^{30} + 152587890625 T^{32}$$
$7$ $$( 1 + 4 T - 4 T^{3} + 29 T^{4} - 28 T^{5} + 1372 T^{7} + 2401 T^{8} )^{2}$$
$11$ $$1 + 34 T^{2} + 631 T^{4} + 7742 T^{6} + 56309 T^{8} + 36244 T^{10} - 4803190 T^{12} - 80571856 T^{14} - 990277910 T^{16} - 9749194576 T^{18} - 70323504790 T^{20} + 64208456884 T^{22} + 12070334230229 T^{24} + 200807541260942 T^{26} + 1980348305710951 T^{28} + 12911494341830194 T^{30} + 45949729863572161 T^{32}$$
$13$ $$( 1 - 22 T^{2} + 597 T^{4} - 10874 T^{6} + 145448 T^{8} - 1837706 T^{10} + 17050917 T^{12} - 106189798 T^{14} + 815730721 T^{16} )^{2}$$
$17$ $$1 - 84 T^{2} + 3336 T^{4} - 99240 T^{6} + 2736514 T^{8} - 66944172 T^{10} + 1418515200 T^{12} - 27661974396 T^{14} + 497038747923 T^{16} - 7994310600444 T^{18} + 118475808019200 T^{20} - 1615869570797868 T^{22} + 19089257897900674 T^{24} - 200067234680558760 T^{26} + 1943627783398482696 T^{28} - 14143737430989678036 T^{30} + 48661191875666868481 T^{32}$$
$19$ $$( 1 - 6 T + 41 T^{2} - 174 T^{3} + 567 T^{4} + 1140 T^{5} - 10940 T^{6} + 109212 T^{7} - 514114 T^{8} + 2075028 T^{9} - 3949340 T^{10} + 7819260 T^{11} + 73892007 T^{12} - 430841226 T^{13} + 1928881121 T^{14} - 5363230434 T^{15} + 16983563041 T^{16} )^{2}$$
$23$ $$1 + 84 T^{2} + 2840 T^{4} + 59304 T^{6} + 1536034 T^{8} + 51810444 T^{10} + 1287398720 T^{12} + 22322505660 T^{14} + 403780671955 T^{16} + 11808605494140 T^{18} + 360266945203520 T^{20} + 7669805137024716 T^{22} + 120288335965115554 T^{24} + 2456757821014240296 T^{26} + 62237533386937711640 T^{28} +$$$$97\!\cdots\!56$$$$T^{30} +$$$$61\!\cdots\!61$$$$T^{32}$$
$29$ $$( 1 - 82 T^{2} + 3369 T^{4} - 137570 T^{6} + 4900772 T^{8} - 115696370 T^{10} + 2382829689 T^{12} - 48775512322 T^{14} + 500246412961 T^{16} )^{2}$$
$31$ $$( 1 - 8 T + 72 T^{2} - 172 T^{3} + 1373 T^{4} - 5332 T^{5} + 69192 T^{6} - 238328 T^{7} + 923521 T^{8} )^{2}( 1 - 4 T + 48 T^{2} - 404 T^{3} + 1502 T^{4} - 12524 T^{5} + 46128 T^{6} - 119164 T^{7} + 923521 T^{8} )^{2}$$
$37$ $$( 1 - 2 T - 39 T^{2} + 686 T^{3} - 133 T^{4} - 26040 T^{5} + 186136 T^{6} + 542152 T^{7} - 8048970 T^{8} + 20059624 T^{9} + 254820184 T^{10} - 1319004120 T^{11} - 249263413 T^{12} + 47569954502 T^{13} - 100063329951 T^{14} - 189863754266 T^{15} + 3512479453921 T^{16} )^{2}$$
$41$ $$( 1 + 204 T^{2} + 20760 T^{4} + 1374660 T^{6} + 65555246 T^{8} + 2310803460 T^{10} + 58662798360 T^{12} + 969021265164 T^{14} + 7984925229121 T^{16} )^{2}$$
$43$ $$( 1 - 2 T + 139 T^{2} - 134 T^{3} + 8158 T^{4} - 5762 T^{5} + 257011 T^{6} - 159014 T^{7} + 3418801 T^{8} )^{4}$$
$47$ $$1 - 200 T^{2} + 20236 T^{4} - 1289200 T^{6} + 54604010 T^{8} - 1402967000 T^{10} + 1590108464 T^{12} + 2082887856200 T^{14} - 134280636932141 T^{16} + 4601099274345800 T^{18} + 7759222059719984 T^{20} - 15122883392481143000 T^{22} +$$$$13\!\cdots\!10$$$$T^{24} -$$$$67\!\cdots\!00$$$$T^{26} +$$$$23\!\cdots\!76$$$$T^{28} -$$$$51\!\cdots\!00$$$$T^{30} +$$$$56\!\cdots\!21$$$$T^{32}$$
$53$ $$1 + 82 T^{2} + 1819 T^{4} - 293986 T^{6} - 25475671 T^{8} - 757240268 T^{10} + 15438451790 T^{12} + 2705625303848 T^{14} + 146204406396370 T^{16} + 7600101478509032 T^{18} + 121816810518410990 T^{20} - 16783746761372742572 T^{22} -$$$$15\!\cdots\!31$$$$T^{24} -$$$$51\!\cdots\!14$$$$T^{26} +$$$$89\!\cdots\!79$$$$T^{28} +$$$$11\!\cdots\!58$$$$T^{30} +$$$$38\!\cdots\!21$$$$T^{32}$$
$59$ $$1 - 322 T^{2} + 52107 T^{4} - 6104798 T^{6} + 600323513 T^{8} - 51577574244 T^{10} + 3916097487742 T^{12} - 266299877019064 T^{14} + 16439030582633874 T^{16} - 926989871903361784 T^{18} + 47452766970162888862 T^{20} -$$$$21\!\cdots\!04$$$$T^{22} +$$$$88\!\cdots\!73$$$$T^{24} -$$$$31\!\cdots\!98$$$$T^{26} +$$$$92\!\cdots\!67$$$$T^{28} -$$$$19\!\cdots\!42$$$$T^{30} +$$$$21\!\cdots\!41$$$$T^{32}$$
$61$ $$( 1 + 70 T^{2} + 4306 T^{4} + 9720 T^{5} - 90560 T^{6} + 641520 T^{7} - 8009105 T^{8} + 39132720 T^{9} - 336973760 T^{10} + 2206255320 T^{11} + 59620191346 T^{12} + 3606426205270 T^{14} + 191707312997281 T^{16} )^{2}$$
$67$ $$( 1 + 14 T - 125 T^{2} - 1238 T^{3} + 28877 T^{4} + 158888 T^{5} - 2519254 T^{6} - 1335236 T^{7} + 243583474 T^{8} - 89460812 T^{9} - 11308931206 T^{10} + 47787631544 T^{11} + 581903921117 T^{12} - 1671454882466 T^{13} - 11307297771125 T^{14} + 84849962474522 T^{15} + 406067677556641 T^{16} )^{2}$$
$71$ $$( 1 - 392 T^{2} + 75892 T^{4} - 9334040 T^{6} + 790096294 T^{8} - 47052895640 T^{10} + 1928543294452 T^{12} - 50215311297032 T^{14} + 645753531245761 T^{16} )^{2}$$
$73$ $$( 1 + 30 T + 591 T^{2} + 8730 T^{3} + 105765 T^{4} + 1168044 T^{5} + 12012906 T^{6} + 115920888 T^{7} + 1041481850 T^{8} + 8462224824 T^{9} + 64016776074 T^{10} + 454388972748 T^{11} + 3003539959365 T^{12} + 18097915006890 T^{13} + 89438527736799 T^{14} + 331421955572910 T^{15} + 806460091894081 T^{16} )^{2}$$
$79$ $$( 1 + 16 T + 64 T^{2} + 56 T^{3} - 1765 T^{4} - 55364 T^{5} - 19696 T^{6} + 3181796 T^{7} + 14045608 T^{8} + 251361884 T^{9} - 122922736 T^{10} - 27296611196 T^{11} - 68746892965 T^{12} + 172315158344 T^{13} + 15557597153344 T^{14} + 307262543778544 T^{15} + 1517108809906561 T^{16} )^{2}$$
$83$ $$( 1 + 438 T^{2} + 94557 T^{4} + 13226610 T^{6} + 1298783528 T^{8} + 91118116290 T^{10} + 4487516458797 T^{12} + 143199883535622 T^{14} + 2252292232139041 T^{16} )^{2}$$
$89$ $$1 - 384 T^{2} + 64644 T^{4} - 8318208 T^{6} + 1128930634 T^{8} - 127988142720 T^{10} + 11460228290064 T^{12} - 1112004852903552 T^{14} + 109920677865950547 T^{16} - 8808190439849035392 T^{18} +$$$$71\!\cdots\!24$$$$T^{20} -$$$$63\!\cdots\!20$$$$T^{22} +$$$$44\!\cdots\!54$$$$T^{24} -$$$$25\!\cdots\!08$$$$T^{26} +$$$$15\!\cdots\!24$$$$T^{28} -$$$$75\!\cdots\!44$$$$T^{30} +$$$$15\!\cdots\!61$$$$T^{32}$$
$97$ $$( 1 - 234 T^{2} + 40305 T^{4} - 5045730 T^{6} + 535964996 T^{8} - 47475273570 T^{10} + 3568172670705 T^{12} - 194915449153386 T^{14} + 7837433594376961 T^{16} )^{2}$$